I grew up with four TV channels, none of which showed anime. However, my older sister had a friend who would bring over tapes of Escaflowne and Gundam Wing, so I was the annoying eightish-year-old sister who would bother them and watch their shows. Our family also rented the first Pokemon movie, which left me kind of confused. I saw Spirited Away when I was about eleven, and I loved it.
When I was thirteen, I got into a vampire phase. (I tried to read several Anne Rice books, but the sex stuff grossed me out.) My sister decided to introduce me to Hellsing, and this was when I realized: 1. There are these cartoons called anime, and 2. I can watch as many as I want for free on the internet. I asked my sister what her favorite anime was, and she said Haibane Renmei. I watched and loved it and I have been into anime ever since. However, I still haven't seen much of Sailor Moon, DBZ, Digimon, Pokemon, Yu-Gi-Oh!, and other staples of the anime experience.